U4AOS1 Topic 2: Lorentz Factor

The Lorentz Factor is the factor by which time and length change for an object in motion.

It can be explained through the Pythagorean Theorem for an understanding of how it works but is not necessary. The important thing is knowing how to calculate the Lorentz Factor and what it means.

The formula for the Lorentz Factor is

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Where $v$ is the velocity in m/s,
and c is the speed of light (or $3*10^8$ m/s).

The Lorentz Factor is easy to calculate. However, it is also easy to make errors when inputting it into the calculator. To reduce the chance of these errors input it by each component.

Below is a graph that shows how the Lorentz Factor changes as the speed is increased

Solving Lorentz Factor
1. Solve for the fraction
$\frac{v^2}{c^2}$
(in most cases, the velocity will be given with respect to c like 0.3c. When this happens just cancel the c terms. Otherwise use the actual value.)
2. Subtract the answer from 1
3. Find the square root using the ANS function
4. Divide 1 by ANS

Calculating the Lorentz Factor by following these steps will ensure that you don't make errors.

Calculating Speed from Lorentz
The process of finding the speed from the Lorentz Factor is very similar
1. Substitute the Lorentz Factor into the equation
2. Swap the Lorentz Factor and the square root
3. Square both sides
4. Subtract 1 on both sides
5. Flip the sign
6. Multiply both sides by $c^2$
7. Take the square root of both sides
8. Depending on how the answer is required make sure to multiply by the value of the speed of light or leave it in terms of c

Example 1

Calculate the Lorentz factor of an object moving at $0.6c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 1: Solve for the fraction component

$\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}$

$0.6^2 = 0.36$

$= 0 . 36$

Step 2: Subtract the result from 1

$1 - 0.36 = 0.64$

Step 3: Calculate the square root

$1 - 0.36 = 0.64$ $\sqrt{0.64} = 0.8$

Step 4: Divide 1 by the result

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Hence the Lorentz Factor is 1.25

Calculate the Lorentz factor of an object moving at $0.6c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 1: Solve for the fraction component

$\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}$

$0.6^2 = 0.36$

$= 0 .9801$

Step 2: Subtract the result from 1

$1 - 0.36 = 0.0199$

Step 3: Calculate the square root

$\sqrt{0.0199} = 0.141$

Step 4: Divide 1 by the result

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Hence the Lorentz Factor is 7.089

Calculate the Lorentz factor of an object moving at $0.2c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 1: Solve for the fraction component

$\frac{v^2}{c^2} = \frac{(0.2c)^2}{c^2}$

$0.2^2 = 0.40$

$= 0 .0 4$

Step 2: Subtract the result from 1

$1 - 0.36 = 0.64$

Step 3: Calculate the square root

$1 - 0.96 = 0.96$ $\sqrt{0.96} = 0.8$

Step 4: Divide 1 by the result

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Hence the Lorentz Factor is 1.02

Example 2

$2.4*10^8$

Calculate the Lorentz factor of an object moving at $0.6c$ $2.4*10^8$ m/s.

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 1: Solve for the fraction component

$\frac{v^2}{c^2} = \frac{2.410^8}{310^8}$

$=\frac{2.4}{3}$

=0.8

$\frac{2.4}{3} = 0.8$

$1 - 0.36 = 0.64$

Step 3: Calculate the square root

$1 - 0.36 = 0.20$ $\sqrt{0.20} = 0.8$

Step 4: Divide 1 by the result

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Hence the Lorentz Factor is 2.236

Example 3

Calculate the speed in terms of $c$ if the Lorentz factor is $15$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 1: Substitute the Lorentz Factor into the equation

$0.2^2 = 0.40$

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Step 2: Swap the Lorentz Factor and the square root

$\sqrt{1-\frac{v^2}{c^2}} = \frac{1}{15}$

Step 3: Square both sides

$1-\frac{v^2}{c^2} = (\frac{1}{15})^2$

$1 - 0.96 = 0.96$

Step 4: Subtract 1 on both sides

$-\frac{v^2}{c^2} = (\frac{1}{15})^2 - 1$

$1 - 0.36 = 0.64$

Step 5: Flip the sign

$\frac{v^2}{c^2} = 1-(\frac{1}{15})^2$

Step 6: Multiply both sides by $c^2$

$v^2 = 0.9955c^2$

Step 7: Take the square root of both sides

$v = 0.9978c$

Hence the velocity is 0.9978c

Exercise &&1&& (&&5&& Questions)

A spaceship is moving at 0.75 $c$ . What is its Lorentz factor?

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A particle is accelerated to 0.9 $c$ . Calculate the Lorentz factor for this particle.

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An alien reports traveling at 0.95 $c$ relative to Earth. What Lorentz factor at this speed?

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A cosmic ray particle is observed traveling at 0.99 $c$ . Find its Lorentz factor.

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An object travels at 0.5 $c$ . What is its corresponding Lorentz factor?

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Exercise &&2&& (&&5&& Questions)

An object has a Lorentz factor of 1.5. At what fraction of the speed of light is it moving? Round to 2 decimals.

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If a particle has a Lorentz factor of 10, calculate its speed in terms of c. Round to 3 decimals.

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An observer notes a spaceship with a Lorentz factor of 4. How fast is this spaceship moving as a fraction of the speed of light? Round to 2 decimals.

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A certain experiment results in particles with a Lorentz factor of 20. Determine their speed as a fraction of $c$ Round to 4 decimals.

Submit

Given a Lorentz factor of 3 for a fast-moving train, find its speed in terms of $c$

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U4AOS1 Topic 2: Lorentz Factor

Calculate the Lorentz factor of an object moving at 0.6c0.6c 0.6c.

﻿We start off by writing the formula

γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}γ=1−c2v2​​1​

﻿

Step 1: Solve for the fraction component

v2c2=(0.6c)2c2\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}c2v2​=c2(0.6c)2​

0.62=0.360.6^2 = 0.36 = 0.62

= 0.36

Step 2: Subtract the result from 1

1−0.36=0.641 - 0.36 = 0.641−0.36=0.64

Step 3: Calculate the square root

1−0.36=0.640.64=0.8\sqrt{0.64} = 0.80.64​=0.8

Step 4: Divide 1 by the result

1−0.36=0.6410.8=1.25\frac{1}{0.8} = 1.250.81​=1.25

Hence the Lorentz Factor is 1.25

Calculate the Lorentz factor of an object moving at 0.6c0.6c 0.99c.

﻿We start off by writing the formula

γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}γ=1−c2v2​​1​

﻿

Step 1: Solve for the fraction component

v2c2=(0.6c)2c2\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}c2v2​=c2(0.99c)2​

0.62=0.360.6^2 = 0.36 = 0.992

= 0.9801

Step 2: Subtract the result from 1

1−0.36=0.641 - 0.36 = 0.01991−0.9801 =0.0199

Step 3: Calculate the square root

​

0.0199=0.141\sqrt{0.0199} = 0.1410.0199​=0.141

Step 4: Divide 1 by the result

1−0.36=0.6410.8=1.25\frac{1}{0.8} = 1.250.1411​= 7.089

Hence the Lorentz Factor is 7.089

Calculate the Lorentz factor of an object moving at 0.2c0.2c 0.2c.

﻿We start off by writing the formula

γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}γ=1−c2v2​​1​

﻿

Step 1: Solve for the fraction component

v2c2=(0.2c)2c2\frac{v^2}{c^2} = \frac{(0.2c)^2}{c^2}c2v2​=c2(0.2c)2​

0.22=0.320.2^2 = 0.40 = 0.22

= 0.04

Step 2: Subtract the result from 1

1−0.36=0.641 - 0.36 = 0.641−0.04=0.96

Step 3: Calculate the square root

1−0.96=0.960.96=0.8\sqrt{0.96} = 0.80.96​=0.98

Step 4: Divide 1 by the result

1−0.36=0.6410.8=1.25\frac{1}{0.8} = 1.250.981​=1.02

Hence the Lorentz Factor is 1.02

2.4∗1082.4*10^8

Calculate the Lorentz factor of an object moving at 0.6c0.6c 2.4∗1082.4*10^82.4∗108 m/s.

﻿We start off by writing the formula

γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}γ=1−c2v2​​1​

﻿

Step 1: Solve for the fraction component

v2c2=2.4∗1083∗108\frac{v^2}{c^2} = \frac{2.4*10^8}{3*10^8}c2v2​=3∗1082.4∗108​

=2.43=\frac{2.4}{3}=32.4​

=0.8

2.43=0.8\frac{2.4}{3} = 0.8Step 2: Subtract the result from 1

1−0.36=0.641 - 0.36 = 0.641−0.8 =0.2

Step 3: Calculate the square root

1−0.36=0.200.20=0.8\sqrt{0.20} = 0.80.20​=0.4472

Step 4: Divide 1 by the result

1−0.36=0.6410.8=1.25\frac{1}{0.8} = 1.250.44721​=2.236

Hence the Lorentz Factor is 2.236

Calculate the speed in terms of ccc if the Lorentz factor is 151515.

﻿We start off by writing the formula

γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}γ=1−c2v2​​1​

Calculate the Lorentz factor of an object moving at $0.6c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

$\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}$

$0.6^2 = 0.36$

$= 0 . 36$

$1 - 0.36 = 0.64$

$1 - 0.36 = 0.64$ $\sqrt{0.64} = 0.8$

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Calculate the Lorentz factor of an object moving at $0.6c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

$\frac{v^2}{c^2} = \frac{(0.6c)^2}{c^2}$

$0.6^2 = 0.36$

$= 0 .9801$

$1 - 0.36 = 0.0199$

$\sqrt{0.0199} = 0.141$

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Calculate the Lorentz factor of an object moving at $0.2c$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

$\frac{v^2}{c^2} = \frac{(0.2c)^2}{c^2}$

$0.2^2 = 0.40$

$= 0 .0 4$

$1 - 0.36 = 0.64$

$1 - 0.96 = 0.96$ $\sqrt{0.96} = 0.8$

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

$2.4*10^8$

Calculate the Lorentz factor of an object moving at $0.6c$ $2.4*10^8$ m/s.

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

$\frac{v^2}{c^2} = \frac{2.410^8}{310^8}$

$=\frac{2.4}{3}$

$\frac{2.4}{3} = 0.8$

$1 - 0.36 = 0.64$

$1 - 0.36 = 0.20$ $\sqrt{0.20} = 0.8$

$1 - 0.36 = 0.64$ $\frac{1}{0.8} = 1.25$

Calculate the speed in terms of $c$ if the Lorentz factor is $15$ .

We start off by writing the formula

$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$